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## 1.7 Summary (EMCF5)

Arithmetic sequence

• common difference $$(d)$$ between any two consecutive terms: $$d = T_{n} - T_{n-1}$$
• general form: $$a + (a + d) + (a + 2d) + \cdots$$
• general formula: $$T_{n} = a + (n - 1)d$$
• graph of the sequence lies on a straight line

• common second difference between any two consecutive terms
• general formula: $$T_{n} = an^{2} + bn + c$$
• graph of the sequence lies on a parabola

Geometric sequence

• constant ratio $$(r)$$ between any two consecutive terms: $$r = \frac{T_{n}}{T_{n-1}}$$
• general form: $$a + ar + ar^{2} + \cdots$$
• general formula: $$T_{n} = ar^{n-1}$$
• graph of the sequence lies on an exponential curve

Sigma notation

$\sum_{k = 1}^{n}{T_{k}}$

Sigma notation is used to indicate the sum of the terms given by $$T_{k}$$, starting from $$k =1$$ and ending at $$k = n$$.

Series

• the sum of certain numbers of terms in a sequence
• arithmetic series:
• $$S_{n} = \frac{n}{2}[a + l]$$
• $$S_{n} = \frac{n}{2}[2a + (n - 1)d]$$
• geometric series:
• $$S_{n} = \frac{a(1 - r^{n})}{1 - r}$$ if $$r < 1$$
• $$S_{n} = \frac{a(r^{n} - 1)}{r-1}$$ if $$r > 1$$

Sum to infinity

A convergent geometric series, with $$- 1 < r < 1$$, tends to a certain fixed number as the number of terms in the sum tends to infinity.

$S_{\infty} = \sum_{n =1}^{\infty}{T_{n}} = \frac{a}{1 - r}$